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Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZAdd.v')
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diff --git a/theories/Numbers/Integer/Abstract/ZAdd.v b/theories/Numbers/Integer/Abstract/ZAdd.v new file mode 100644 index 0000000000..c4c5174dac --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZAdd.v @@ -0,0 +1,299 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) +(* Evgeny Makarov, INRIA, 2007 *) +(************************************************************************) + +Require Export ZBase. + +Module ZAddProp (Import Z : ZAxiomsMiniSig'). +Include ZBaseProp Z. + +(** Theorems that are either not valid on N or have different proofs + on N and Z *) + +Hint Rewrite opp_0 : nz. + +Theorem add_pred_l : forall n m, P n + m == P (n + m). +Proof. +intros n m. +rewrite <- (succ_pred n) at 2. +now rewrite add_succ_l, pred_succ. +Qed. + +Theorem add_pred_r : forall n m, n + P m == P (n + m). +Proof. +intros n m; rewrite 2 (add_comm n); apply add_pred_l. +Qed. + +Theorem add_opp_r : forall n m, n + (- m) == n - m. +Proof. +nzinduct m. +now nzsimpl. +intro m. rewrite opp_succ, sub_succ_r, add_pred_r. now rewrite pred_inj_wd. +Qed. + +Theorem sub_0_l : forall n, 0 - n == - n. +Proof. +intro n; rewrite <- add_opp_r; now rewrite add_0_l. +Qed. + +Theorem sub_succ_l : forall n m, S n - m == S (n - m). +Proof. +intros n m; rewrite <- 2 add_opp_r; now rewrite add_succ_l. +Qed. + +Theorem sub_pred_l : forall n m, P n - m == P (n - m). +Proof. +intros n m. rewrite <- (succ_pred n) at 2. +rewrite sub_succ_l; now rewrite pred_succ. +Qed. + +Theorem sub_pred_r : forall n m, n - (P m) == S (n - m). +Proof. +intros n m. rewrite <- (succ_pred m) at 2. +rewrite sub_succ_r; now rewrite succ_pred. +Qed. + +Theorem opp_pred : forall n, - (P n) == S (- n). +Proof. +intro n. rewrite <- (succ_pred n) at 2. +rewrite opp_succ. now rewrite succ_pred. +Qed. + +Theorem sub_diag : forall n, n - n == 0. +Proof. +nzinduct n. +now nzsimpl. +intro n. rewrite sub_succ_r, sub_succ_l; now rewrite pred_succ. +Qed. + +Theorem add_opp_diag_l : forall n, - n + n == 0. +Proof. +intro n; now rewrite add_comm, add_opp_r, sub_diag. +Qed. + +Theorem add_opp_diag_r : forall n, n + (- n) == 0. +Proof. +intro n; rewrite add_comm; apply add_opp_diag_l. +Qed. + +Theorem add_opp_l : forall n m, - m + n == n - m. +Proof. +intros n m; rewrite <- add_opp_r; now rewrite add_comm. +Qed. + +Theorem add_sub_assoc : forall n m p, n + (m - p) == (n + m) - p. +Proof. +intros n m p; rewrite <- 2 add_opp_r; now rewrite add_assoc. +Qed. + +Theorem opp_involutive : forall n, - (- n) == n. +Proof. +nzinduct n. +now nzsimpl. +intro n. rewrite opp_succ, opp_pred. now rewrite succ_inj_wd. +Qed. + +Theorem opp_add_distr : forall n m, - (n + m) == - n + (- m). +Proof. +intros n m; nzinduct n. +now nzsimpl. +intro n. rewrite add_succ_l; do 2 rewrite opp_succ; rewrite add_pred_l. +now rewrite pred_inj_wd. +Qed. + +Theorem opp_sub_distr : forall n m, - (n - m) == - n + m. +Proof. +intros n m; rewrite <- add_opp_r, opp_add_distr. +now rewrite opp_involutive. +Qed. + +Theorem opp_inj : forall n m, - n == - m -> n == m. +Proof. +intros n m H. apply opp_wd in H. now rewrite 2 opp_involutive in H. +Qed. + +Theorem opp_inj_wd : forall n m, - n == - m <-> n == m. +Proof. +intros n m; split; [apply opp_inj | intros; now f_equiv]. +Qed. + +Theorem eq_opp_l : forall n m, - n == m <-> n == - m. +Proof. +intros n m. now rewrite <- (opp_inj_wd (- n) m), opp_involutive. +Qed. + +Theorem eq_opp_r : forall n m, n == - m <-> - n == m. +Proof. +symmetry; apply eq_opp_l. +Qed. + +Theorem sub_add_distr : forall n m p, n - (m + p) == (n - m) - p. +Proof. +intros n m p; rewrite <- add_opp_r, opp_add_distr, add_assoc. +now rewrite 2 add_opp_r. +Qed. + +Theorem sub_sub_distr : forall n m p, n - (m - p) == (n - m) + p. +Proof. +intros n m p; rewrite <- add_opp_r, opp_sub_distr, add_assoc. +now rewrite add_opp_r. +Qed. + +Theorem sub_opp_l : forall n m, - n - m == - m - n. +Proof. +intros n m. rewrite <- 2 add_opp_r. now rewrite add_comm. +Qed. + +Theorem sub_opp_r : forall n m, n - (- m) == n + m. +Proof. +intros n m; rewrite <- add_opp_r; now rewrite opp_involutive. +Qed. + +Theorem add_sub_swap : forall n m p, n + m - p == n - p + m. +Proof. +intros n m p. rewrite <- add_sub_assoc, <- (add_opp_r n p), <- add_assoc. +now rewrite add_opp_l. +Qed. + +Theorem sub_cancel_l : forall n m p, n - m == n - p <-> m == p. +Proof. +intros n m p. rewrite <- (add_cancel_l (n - m) (n - p) (- n)). +rewrite 2 add_sub_assoc. rewrite add_opp_diag_l; rewrite 2 sub_0_l. +apply opp_inj_wd. +Qed. + +Theorem sub_cancel_r : forall n m p, n - p == m - p <-> n == m. +Proof. +intros n m p. +stepl (n - p + p == m - p + p) by apply add_cancel_r. +now do 2 rewrite <- sub_sub_distr, sub_diag, sub_0_r. +Qed. + +(** The next several theorems are devoted to moving terms from one + side of an equation to the other. The name contains the operation + in the original equation ([add] or [sub]) and the indication + whether the left or right term is moved. *) + +Theorem add_move_l : forall n m p, n + m == p <-> m == p - n. +Proof. +intros n m p. +stepl (n + m - n == p - n) by apply sub_cancel_r. +now rewrite add_comm, <- add_sub_assoc, sub_diag, add_0_r. +Qed. + +Theorem add_move_r : forall n m p, n + m == p <-> n == p - m. +Proof. +intros n m p; rewrite add_comm; now apply add_move_l. +Qed. + +(** The two theorems above do not allow rewriting subformulas of the + form [n - m == p] to [n == p + m] since subtraction is in the + right-hand side of the equation. Hence the following two + theorems. *) + +Theorem sub_move_l : forall n m p, n - m == p <-> - m == p - n. +Proof. +intros n m p; rewrite <- (add_opp_r n m); apply add_move_l. +Qed. + +Theorem sub_move_r : forall n m p, n - m == p <-> n == p + m. +Proof. +intros n m p; rewrite <- (add_opp_r n m). now rewrite add_move_r, sub_opp_r. +Qed. + +Theorem add_move_0_l : forall n m, n + m == 0 <-> m == - n. +Proof. +intros n m; now rewrite add_move_l, sub_0_l. +Qed. + +Theorem add_move_0_r : forall n m, n + m == 0 <-> n == - m. +Proof. +intros n m; now rewrite add_move_r, sub_0_l. +Qed. + +Theorem sub_move_0_l : forall n m, n - m == 0 <-> - m == - n. +Proof. +intros n m. now rewrite sub_move_l, sub_0_l. +Qed. + +Theorem sub_move_0_r : forall n m, n - m == 0 <-> n == m. +Proof. +intros n m. now rewrite sub_move_r, add_0_l. +Qed. + +(** The following section is devoted to cancellation of like + terms. The name includes the first operator and the position of + the term being canceled. *) + +Theorem add_simpl_l : forall n m, n + m - n == m. +Proof. +intros; now rewrite add_sub_swap, sub_diag, add_0_l. +Qed. + +Theorem add_simpl_r : forall n m, n + m - m == n. +Proof. +intros; now rewrite <- add_sub_assoc, sub_diag, add_0_r. +Qed. + +Theorem sub_simpl_l : forall n m, - n - m + n == - m. +Proof. +intros; now rewrite <- add_sub_swap, add_opp_diag_l, sub_0_l. +Qed. + +Theorem sub_simpl_r : forall n m, n - m + m == n. +Proof. +intros; now rewrite <- sub_sub_distr, sub_diag, sub_0_r. +Qed. + +Theorem sub_add : forall n m, m - n + n == m. +Proof. + intros. now rewrite <- add_sub_swap, add_simpl_r. +Qed. + +(** Now we have two sums or differences; the name includes the two + operators and the position of the terms being canceled *) + +Theorem add_add_simpl_l_l : forall n m p, (n + m) - (n + p) == m - p. +Proof. +intros n m p. now rewrite (add_comm n m), <- add_sub_assoc, +sub_add_distr, sub_diag, sub_0_l, add_opp_r. +Qed. + +Theorem add_add_simpl_l_r : forall n m p, (n + m) - (p + n) == m - p. +Proof. +intros n m p. rewrite (add_comm p n); apply add_add_simpl_l_l. +Qed. + +Theorem add_add_simpl_r_l : forall n m p, (n + m) - (m + p) == n - p. +Proof. +intros n m p. rewrite (add_comm n m); apply add_add_simpl_l_l. +Qed. + +Theorem add_add_simpl_r_r : forall n m p, (n + m) - (p + m) == n - p. +Proof. +intros n m p. rewrite (add_comm p m); apply add_add_simpl_r_l. +Qed. + +Theorem sub_add_simpl_r_l : forall n m p, (n - m) + (m + p) == n + p. +Proof. +intros n m p. now rewrite <- sub_sub_distr, sub_add_distr, sub_diag, +sub_0_l, sub_opp_r. +Qed. + +Theorem sub_add_simpl_r_r : forall n m p, (n - m) + (p + m) == n + p. +Proof. +intros n m p. rewrite (add_comm p m); apply sub_add_simpl_r_l. +Qed. + +(** Of course, there are many other variants *) + +End ZAddProp. + |